Kinetic description of optimal control problems and applications to opinion consensus

Giacomo Albi, Michael Herty, Lorenzo Pareschi
(30/01/2014 Communications in Mathematical Sciences, 13 (2015)
1407 – 1429, arXiv:1401.7798)

In this paper an optimal control problem for a large system of interacting agents is considered using a kinetic perspective. As a prototype model we analyze a microscopic model of opinion formation under constraints. For this problem a Boltzmann-type equation based on a model predictive control formulation is introduced and discussed.

In particular, the receding horizon  strategy permits to embed the minimization of suitable cost functional into  binary particle interactions. The corresponding Fokker-Planck asymptotic limit is also derived and explicit expressions of stationary solutions are given. Several numerical results showing the robustness of the present approach are finally reported.